\(\int \frac {x^2}{(a+b x^2+c x^4)^3} \, dx\) [814]

Optimal result

Integrand size = 18, antiderivative size = 311 \[ \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx=-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {x \left (b \left (b^2+8 a c\right )+c \left (b^2+20 a c\right ) x^2\right )}{8 a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} \left (b^2+20 a c+\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (b^2+20 a c-\frac {b \left (b^2-52 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{8 \sqrt {2} a \left (b^2-4 a c\right )^2 \sqrt {b+\sqrt {b^2-4 a c}}} \] Output:

-1/4*x*(2*c*x^2+b)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^2+1/8*x*(b*(8*a*c+b^2)+c*( 
20*a*c+b^2)*x^2)/a/(-4*a*c+b^2)^2/(c*x^4+b*x^2+a)+1/16*c^(1/2)*(b^2+20*a*c 
+b*(-52*a*c+b^2)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b 
^2)^(1/2))^(1/2))*2^(1/2)/a/(-4*a*c+b^2)^2/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/ 
16*c^(1/2)*(b^2+20*a*c-b*(-52*a*c+b^2)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)* 
c^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a/(-4*a*c+b^2)^2/(b+(-4*a* 
c+b^2)^(1/2))^(1/2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.07 \[ \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx=\frac {1}{16} \left (-\frac {4 x \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {2 x \left (b^3+8 a b c+b^2 c x^2+20 a c^2 x^2\right )}{a \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \sqrt {c} \left (b^3-52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-b^3+52 a b c+b^2 \sqrt {b^2-4 a c}+20 a c \sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}\right ) \] Input:

Integrate[x^2/(a + b*x^2 + c*x^4)^3,x]
 

Output:

((-4*x*(b + 2*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + (2*x*(b^3 + 
8*a*b*c + b^2*c*x^2 + 20*a*c^2*x^2))/(a*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4 
)) + (Sqrt[2]*Sqrt[c]*(b^3 - 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqr 
t[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/( 
a*(b^2 - 4*a*c)^(5/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-b^ 
3 + 52*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 20*a*c*Sqrt[b^2 - 4*a*c])*ArcTan[(S 
qrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(5/2)*Sqr 
t[b + Sqrt[b^2 - 4*a*c]]))/16
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {1439, 1492, 25, 1480, 218}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1439

\(\displaystyle \frac {\int \frac {b-10 c x^2}{\left (c x^4+b x^2+a\right )^2}dx}{4 \left (b^2-4 a c\right )}-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

\(\Big \downarrow \) 1492

\(\displaystyle \frac {\frac {x \left (c x^2 \left (20 a c+b^2\right )+b \left (8 a c+b^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {c \left (b^2+20 a c\right ) x^2+b \left (b^2-16 a c\right )}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}}{4 \left (b^2-4 a c\right )}-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {c \left (b^2+20 a c\right ) x^2+b \left (b^2-16 a c\right )}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a c+b^2\right )+b \left (8 a c+b^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {\frac {1}{2} c \left (-\frac {52 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} c \left (\frac {52 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a c+b^2\right )+b \left (8 a c+b^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {\sqrt {c} \left (-\frac {52 a b c}{\sqrt {b^2-4 a c}}+\frac {b^3}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {c} \left (\frac {52 a b c}{\sqrt {b^2-4 a c}}-\frac {b^3}{\sqrt {b^2-4 a c}}+20 a c+b^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c x^2 \left (20 a c+b^2\right )+b \left (8 a c+b^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}}{4 \left (b^2-4 a c\right )}-\frac {x \left (b+2 c x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}\)

Input:

Int[x^2/(a + b*x^2 + c*x^4)^3,x]
 

Output:

-1/4*(x*(b + 2*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) + ((x*(b*(b^2 
 + 8*a*c) + c*(b^2 + 20*a*c)*x^2))/(2*a*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) 
 + ((Sqrt[c]*(b^2 + 20*a*c + b^3/Sqrt[b^2 - 4*a*c] - (52*a*b*c)/Sqrt[b^2 - 
 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2] 
*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[c]*(b^2 + 20*a*c - b^3/Sqrt[b^2 - 4* 
a*c] + (52*a*b*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + S 
qrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a*(b^2 - 4*a 
*c)))/(4*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1439

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[d*(d*x)^(m - 1)*(b + 2*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*(p + 
1)*(b^2 - 4*a*c))), x] - Simp[d^2/(2*(p + 1)*(b^2 - 4*a*c))   Int[(d*x)^(m 
- 2)*(b*(m - 1) + 2*c*(m + 4*p + 5)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x 
] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 
1] && LeQ[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

rule 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + 
 c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 
 - 4*a*c))   Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 LtQ[p, -1] && IntegerQ[2*p]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.93

method	result	size
risch	\(\frac {\frac {c^{2} \left (20 a c +b^{2}\right ) x^{7}}{8 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {c b \left (14 a c +b^{2}\right ) x^{5}}{4 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (36 a^{2} c^{2}+5 a \,b^{2} c +b^{4}\right ) x^{3}}{8 a \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {b \left (16 a c -b^{2}\right ) x}{128 a^{2} c^{2}-64 a \,b^{2} c +8 b^{4}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} c +\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {c \left (20 a c +b^{2}\right ) \textit {\_R}^{2}}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}-\frac {b \left (16 a c -b^{2}\right )}{16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}}{16 a}\)	\(289\)
default	\(64 c^{3} \left (-\frac {\frac {-\frac {\left (320 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}-144 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}+12 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c +\sqrt {-4 a c +b^{2}}\, b^{6}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) x^{3}}{16 a \,c^{2}}+\frac {\left (-96 \sqrt {-4 a c +b^{2}}\, a^{2} b \,c^{2}+48 \sqrt {-4 a c +b^{2}}\, a \,b^{3} c -6 \sqrt {-4 a c +b^{2}}\, b^{5}+448 a^{3} c^{3}-336 a^{2} b^{2} c^{2}+84 a \,b^{4} c -7 b^{6}\right ) x}{8 c^{2}}}{\left (x^{2}+\frac {\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}\right )^{2}}-\frac {\left (320 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}-144 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}+12 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c +\sqrt {-4 a c +b^{2}}\, b^{6}+832 a^{3} b \,c^{3}-432 a^{2} b^{3} c^{2}+60 a \,b^{5} c -b^{7}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{64 c \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \left (4 a c -b^{2}\right )^{2}}+\frac {\frac {-\frac {\left (-320 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}+144 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}-12 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c -\sqrt {-4 a c +b^{2}}\, b^{6}+64 a^{3} b \,c^{3}-48 a^{2} b^{3} c^{2}+12 a \,b^{5} c -b^{7}\right ) x^{3}}{16 a \,c^{2}}+\frac {\left (96 \sqrt {-4 a c +b^{2}}\, a^{2} b \,c^{2}-48 \sqrt {-4 a c +b^{2}}\, a \,b^{3} c +6 \sqrt {-4 a c +b^{2}}\, b^{5}+448 a^{3} c^{3}-336 a^{2} b^{2} c^{2}+84 a \,b^{4} c -7 b^{6}\right ) x}{8 c^{2}}}{\left (x^{2}+\frac {b}{2 c}-\frac {\sqrt {-4 a c +b^{2}}}{2 c}\right )^{2}}-\frac {\left (320 \sqrt {-4 a c +b^{2}}\, a^{3} c^{3}-144 \sqrt {-4 a c +b^{2}}\, a^{2} b^{2} c^{2}+12 \sqrt {-4 a c +b^{2}}\, a \,b^{4} c +\sqrt {-4 a c +b^{2}}\, b^{6}-832 a^{3} b \,c^{3}+432 a^{2} b^{3} c^{2}-60 a \,b^{5} c +b^{7}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{64 c \left (-4 a c +b^{2}\right )^{\frac {5}{2}} \left (4 a c -b^{2}\right )^{2}}\right )\)	\(836\)

Input:

int(x^2/(c*x^4+b*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

(1/8*c^2*(20*a*c+b^2)/a/(16*a^2*c^2-8*a*b^2*c+b^4)*x^7+1/4*c/a*b*(14*a*c+b 
^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^5+1/8*(36*a^2*c^2+5*a*b^2*c+b^4)/a/(16*a^ 
2*c^2-8*a*b^2*c+b^4)*x^3+1/8*b*(16*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x)/ 
(c*x^4+b*x^2+a)^2+1/16/a*sum((c*(20*a*c+b^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*_R 
^2-b*(16*a*c-b^2)/(16*a^2*c^2-8*a*b^2*c+b^4))/(2*_R^3*c+_R*b)*ln(x-_R),_R= 
RootOf(_Z^4*c+_Z^2*b+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3777 vs. \(2 (267) = 534\).

Time = 0.30 (sec) , antiderivative size = 3777, normalized size of antiderivative = 12.14 \[ \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^2/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate(x**2/(c*x**4+b*x**2+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx=\int { \frac {x^{2}}{{\left (c x^{4} + b x^{2} + a\right )}^{3}} \,d x } \] Input:

integrate(x^2/(c*x^4+b*x^2+a)^3,x, algorithm="maxima")
 

Output:

1/8*((b^2*c^2 + 20*a*c^3)*x^7 + 2*(b^3*c + 14*a*b*c^2)*x^5 + (b^4 + 5*a*b^ 
2*c + 36*a^2*c^2)*x^3 - (a*b^3 - 16*a^2*b*c)*x)/((a*b^4*c^2 - 8*a^2*b^2*c^ 
3 + 16*a^3*c^4)*x^8 + a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2 + 2*(a*b^5*c - 8* 
a^2*b^3*c^2 + 16*a^3*b*c^3)*x^6 + (a*b^6 - 6*a^2*b^4*c + 32*a^4*c^3)*x^4 + 
 2*(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2) + 1/8*integrate((b^3 - 16*a 
*b*c + (b^2*c + 20*a*c^2)*x^2)/(c*x^4 + b*x^2 + a), x)/(a*b^4 - 8*a^2*b^2* 
c + 16*a^3*c^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4270 vs. \(2 (267) = 534\).

Time = 1.42 (sec) , antiderivative size = 4270, normalized size of antiderivative = 13.73 \[ \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate(x^2/(c*x^4+b*x^2+a)^3,x, algorithm="giac")
 

Output:

1/64*(2*a^2*b^12*c^2 - 136*a^3*b^10*c^3 + 1856*a^4*b^8*c^4 - 10496*a^5*b^6 
*c^5 + 27136*a^6*b^4*c^6 - 26624*a^7*b^2*c^7 - sqrt(2)*sqrt(b^2 - 4*a*c)*s 
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^12 + 68*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr 
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^10*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt 
(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^11*c - 928*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr 
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^8*c^2 - 128*sqrt(2)*sqrt(b^2 - 4*a*c)*s 
qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^9*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqr 
t(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^10*c^2 + 5248*sqrt(2)*sqrt(b^2 - 4*a*c) 
*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^6*c^3 + 1344*sqrt(2)*sqrt(b^2 - 4*a 
*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^7*c^3 + 64*sqrt(2)*sqrt(b^2 - 4* 
a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*b^8*c^3 - 13568*sqrt(2)*sqrt(b^2 
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^4*c^4 - 5120*sqrt(2)*sqrt(b 
^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^5*c^4 - 672*sqrt(2)*sqrt 
(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^6*c^4 + 13312*sqrt(2)* 
sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^2*c^5 + 6656*sqrt( 
2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^3*c^5 + 2560*sq 
rt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^4*c^5 - 3328 
*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^2*c^6 - 2 
*(b^2 - 4*a*c)*a^2*b^10*c^2 + 128*(b^2 - 4*a*c)*a^3*b^8*c^3 - 1344*(b^2 - 
4*a*c)*a^4*b^6*c^4 + 5120*(b^2 - 4*a*c)*a^5*b^4*c^5 - 6656*(b^2 - 4*a*c...
 

Mupad [B] (verification not implemented)

Time = 22.54 (sec) , antiderivative size = 9731, normalized size of antiderivative = 31.29 \[ \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

int(x^2/(a + b*x^2 + c*x^4)^3,x)
 

Output:

((b*x*(16*a*c - b^2))/(8*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (x^3*(b^4 + 36* 
a^2*c^2 + 5*a*b^2*c))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (b*x^5*(14*a* 
c^2 + b^2*c))/(4*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (c*x^7*(20*a*c^2 + b^ 
2*c))/(8*a*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^4*(2*a*c + b^2) + a^2 + c^2 
*x^8 + 2*a*b*x^2 + 2*b*c*x^6) + atan(((((256*a*b^13*c^2 + 4194304*a^7*b*c^ 
8 - 9216*a^2*b^11*c^3 + 122880*a^3*b^9*c^4 - 819200*a^4*b^7*c^5 + 2949120* 
a^5*b^5*c^6 - 5505024*a^6*b^3*c^7)/(512*(a^2*b^12 + 4096*a^8*c^6 - 24*a^3* 
b^10*c + 240*a^4*b^8*c^2 - 1280*a^5*b^6*c^3 + 3840*a^6*b^4*c^4 - 6144*a^7* 
b^2*c^5)) - (x*(-(b^17 + b^2*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c^8 
 + 1140*a^2*b^13*c^2 - 10160*a^3*b^11*c^3 + 34880*a^4*b^9*c^4 + 43776*a^5* 
b^7*c^5 - 680960*a^6*b^5*c^6 + 1863680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a*c* 
(-(4*a*c - b^2)^15)^(1/2))/(512*(a^3*b^20 + 1048576*a^13*c^10 - 40*a^4*b^1 
8*c + 720*a^5*b^16*c^2 - 7680*a^6*b^14*c^3 + 53760*a^7*b^12*c^4 - 258048*a 
^8*b^10*c^5 + 860160*a^9*b^8*c^6 - 1966080*a^10*b^6*c^7 + 2949120*a^11*b^4 
*c^8 - 2621440*a^12*b^2*c^9)))^(1/2)*(262144*a^7*b*c^7 - 256*a^2*b^11*c^2 
+ 5120*a^3*b^9*c^3 - 40960*a^4*b^7*c^4 + 163840*a^5*b^5*c^5 - 327680*a^6*b 
^3*c^6))/(32*(a^2*b^8 + 256*a^6*c^4 - 16*a^3*b^6*c + 96*a^4*b^4*c^2 - 256* 
a^5*b^2*c^3)))*(-(b^17 + b^2*(-(4*a*c - b^2)^15)^(1/2) - 1720320*a^8*b*c^8 
 + 1140*a^2*b^13*c^2 - 10160*a^3*b^11*c^3 + 34880*a^4*b^9*c^4 + 43776*a^5* 
b^7*c^5 - 680960*a^6*b^5*c^6 + 1863680*a^7*b^3*c^7 - 55*a*b^15*c - 25*a...
 

Reduce [B] (verification not implemented)

Time = 4.44 (sec) , antiderivative size = 5719, normalized size of antiderivative = 18.39 \[ \int \frac {x^2}{\left (a+b x^2+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

int(x^2/(c*x^4+b*x^2+a)^3,x)
 

Output:

( - 80*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - 
b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*c**2 - 36*sqrt(a)*sqrt 
(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/s 
qrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*c - 160*sqrt(a)*sqrt(2*sqrt(c)*sqrt( 
a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sq 
rt(a) + b))*a**3*b*c**2*x**2 - 160*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*ata 
n((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b)) 
*a**3*c**3*x**4 + 2*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt( 
c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**4 - 72 
*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2 
*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**3*c*x**2 - 152*sqrt(a)*sq 
rt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x) 
/sqrt(2*sqrt(c)*sqrt(a) + b))*a**2*b**2*c**2*x**4 - 160*sqrt(a)*sqrt(2*sqr 
t(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2* 
sqrt(c)*sqrt(a) + b))*a**2*b*c**3*x**6 - 80*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) 
 + b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt 
(a) + b))*a**2*c**4*x**8 + 4*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqr 
t(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b** 
5*x**2 - 32*sqrt(a)*sqrt(2*sqrt(c)*sqrt(a) + b)*atan((sqrt(2*sqrt(c)*sqrt( 
a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a*b**4*c*x**4 - 72*...